give a, b and c are rational numbers, show the expressions are rational.
i.) a^2 + b^2 + c ^2
ii.) (5a^3 +6b^4)/(a^2+b^2)
iii.) a + a^2 + a^3 + ... +a^10
I had alot of trouble proving this, and it was not enough work.
please explain these.
give a, b and c are rational numbers, show the expressions are rational.
i.) a^2 + b^2 + c ^2
ii.) (5a^3 +6b^4)/(a^2+b^2)
iii.) a + a^2 + a^3 + ... +a^10
I had alot of trouble proving this, and it was not enough work.
please explain these.
Write a, b and c as ratio of integers, then substitute these into the expressions and rearrange to show that they are the ratios of integers.
For example let $\displaystyle a=e/f$, $\displaystyle b=g/h$, $\displaystyle c=i/j$, where $\displaystyle e,\ f,\ g,\ i$ and $\displaystyle j$ are integers.
Then:
$\displaystyle a^2 + b^2 + c ^2=\frac{e^2}{f^2}+\frac{g^2}{h^2}+\frac{i^2}{j^2} =\frac{e^2h^2j^2+g^2f^2j^2+i^2f^2h^2}{f^2h^2j^2}$
RonL
Speaking generally there are a lot of different operations one can do to rational numbers without leaving the rational world.
The product of any finite number of rational numbers is rational:
let $\displaystyle q_1, q_2, ... q_k $ be rational numbers.
say $\displaystyle q_i=\frac{n_i}{m_i}, \forall i, \text{ and } m_i, n_i $ are integers.
then $\displaystyle n_1n_2...n_k \text{ and } m_1m_2...m_k $ are integers
therefore $\displaystyle q_1 q_2 ... q_k $ is rational.
Same kind of proofs can be used to show the sum of 2 rationals is rational, which extends inductively to the finite sum of rationals. The division of rationals remaining rational is a direct consequence of the multiplication of rationals remaining rational. So it is not hard to see that any finite composition of such operations on rationals results in a rational. Proving these general statements might be easier than specific examples which can get very messy.